In this paper a theorem on the absolute Nörlund summability factors has been proved under more weaker conditions by using a quasiβ-power increasing sequence instead of an almost increasing sequence

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Volume 6, Issue 3, Article 62, 2005

ABSOLUTE NÖRLUND SUMMABILITY FACTORS

HÜSEY˙IN BOR

DEPARTMENT OFMATHEMATICS

ERCIYESUNIVERSITY

38039 KAYSERI, TURKEY

bor@erciyes.edu.tr

URL:http://fef.erciyes.edu.tr/math/hbor.htm

Received 24 May, 2005; accepted 31 May, 2005 Communicated by L. Leindler

ABSTRACT. In this paper a theorem on the absolute Nörlund summability factors has been proved under more weaker conditions by using a quasiβ-power increasing sequence instead of an almost increasing sequence.

Key words and phrases: Nörlund summability, summability factors, power increasing sequences.

2000 Mathematics Subject Classification. 40D15, 40F05, 40G05.

1. INTRODUCTION

A positive sequence (b_n)is said to be almost increasing if there exist a positive increasing sequence(c_n)and two positive constantsAandBsuch thatAc_n≤b_n≤Bc_n(see [2]).

LetP

a_nbe a given infinite series with the sequence of partial sums(s_n)andw_n =na_n. By u^α_nandt^α_n we denote then-th Cesàro means of orderα, withα >−1, of the sequences(s_n)and (wn), respectively. The seriesP

anis said to be summable|C, α|, if (see [5], [7]) (1.1)

∞

X

n=1

u^α_n−u^α_n−1 =

∞

X

n=1

1

n|t^α_n|<∞.

Let(pn)be a sequence of constants, real or complex, and let us write (1.2) Pn=p0 +p1+p2+· · ·+pn6= 0, (n ≥0).

The sequence-to-sequence transformation

(1.3) σ_n= 1

P_n

n

X

v=0

p_n−vs_v

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

163-05

defines the sequence(σ_n)of the Nörlund mean of the sequence(s_n), generated by the sequence of coefficients(p_n). The seriesP

a_nis said to be summable|N, p_n|, if (see [9]) (1.4)

∞

X

n=1

|σ_n−σn−1|<∞.

In the special case when

(1.5) p_n = Γ(n+α)

Γ(α)Γ(n+ 1), α≥0

the Nörlund mean reduces to the(C, α)mean and|N, p_n|summability becomes|C, α|summa- bility. Forp_n = 1andP_n = n, we get the(C,1)mean and then|N, p_n|summability becomes

|C,1|summability. For any sequence(λ_n), we write∆λ_n=λ_n−λ_n+1and∆²λ_n= ∆(∆λ_n) =

∆λ_n−∆λ_n+1.

In [6] Kishore has proved the following theorem concerning|C,1|and|N, p_n|summability methods.

Theorem 1.1. Letp₀ >0,p_n ≥0and(p_n)be a non-increasing sequence. IfP

a_nis summable

|C,1|, then the seriesP

a_nP_n(n+ 1)⁻¹ is summable|N, p_n|.

Ahmad [1] proved the following theorem for absolute Nörlund summability factors.

Theorem 1.2. Let(p_n)be as in Theorem 1.1. If (1.6)

n

X

v=1

1

v |t_v|=O(X_n) asn→ ∞,

where(X_n)is a positive non-decreasing sequence and(λ_n)is a sequence such that

(1.7) X_nλ_n =O(1),

(1.8) n∆X_n =O(X_n),

(1.9) X

nXn

∆²λn

<∞, then the seriesP

a_nP_nλ_n(n+ 1)⁻¹is summable|N, p_n|.

Later on Bor [3] proved Theorem 1.2 under weaker conditions in the following form.

Theorem 1.3. Let (p_n) be as in Theorem 1.1 and let (X_n) be a positive non-decreasing se- quence. If the conditions (1.6) and (1.7) of Theorem 1.2 are satisfied and the sequences (λ_n) and(β_n)are such that

(1.10) |∆λ_n| ≤β_n,

(1.11) β_n →0,

(1.12) X

nX_n|∆β_n|<∞, then the seriesP

anPnλn(n+ 1)⁻¹is summable|N, pn|.

Also Bor [4] has proved Theorem 1.3 under the weaker conditions in the following form.

Theorem 1.4. Let(p_n)be as in Theorem 1.1 and let(X_n)be an almost increasing sequence. If the conditions (1.6), (1.7), (1.10) and (1.12) of Theorem 1.2 and Theorem 1.3 are satisfied, then the seriesP

a_nP_nλ_n(n+ 1)⁻¹ is summable|N, p_n|.

2. MAINRESULT

The aim of this paper is to prove Theorem 1.4 under more weaker conditions. For this we need the concept of a quasiβ-power increasing sequence. A positive sequence(γ_n)is said to be a quasiβ-power increasing sequence if there exists a constantK =K(β, γ)≥1such that

(2.1) Kn^βγ_n≥m^βγ_m

holds for all n ≥ m ≥ 1. It should be noted that every almost increasing sequence is a quasi β-power increasing sequence for any nonnegativeβ, but the converse need not be true as can be seen by taking an example, say γ_n =n^−β for β > 0. So we are weakening the hypotheses of the theorem replacing an almost increasing sequence by a quasiβ-power increasing sequence.

Now we shall prove the following theorem.

Theorem 2.1. Let (p_n) be as in Theorem 1.1 and let (X_n) be a quasi β-power increasing sequence. If the conditions (1.6), (1.7), (1.10) and (1.12) of Theorem 1.2 and Theorem 1.3 are satisfied, then the seriesP

a_nP_nλ_n(n+ 1)⁻¹ is summable|N, p_n|.

We need the following lemma for the proof of our theorem.

Lemma 2.2 ([8]). Under the conditions on(X_n), (λ_n)and(β_n), as taken in the statement of the theorem, the following conditions hold, when (1.12) is satisfied:

(2.2) nβ_nX_n=O(1) asn→ ∞,

(2.3)

∞

X

n=1

β_nX_n<∞.

Proof of Theorem 2.1. In order to prove the theorem, we need consider only the special case in which(N, p_n)is (C,1), that is, we shall prove thatP

a_nλ_n is summable|C,1|. Our theorem will then follow by means of Theorem 1.1. Let Tn be the n-th (C,1) mean of the sequence (nanλn), that is,

(2.4) Tn= 1

n+ 1

n

X

v=1

vavλv.

Using Abel’s transformation, we have T_n= 1

n+ 1

n

X

v=1

va_vλ_v

= 1

n+ 1

n−1

X

v=1

∆λ_v(v + 1)t_v +λ_nt_n

=T_n,1+T_n,2, say.

To complete the proof of the theorem, it is sufficient to show that (2.5)

∞

X

n=1

1

n|T_n,r|<∞ forr = 1,2, by (1.1).

Now, we have

m+1

X

n=2

1

n|T_n,1| ≤

m+1

X

n=2

1 n(n+ 1)

(_n−1 X

v=1

v + 1

v v|∆λ_v| |t_v| )

=O(1)

m+1

X

n=2

1 n²

(_n−1 X

v=1

vβ_v|t_v| )

=O(1)

m

X

v=1

vβ_v|t_v|

m+1

X

n=v+1

1 n²

=O(1)

m

X

v=1

vβ_v|tv| v

=O(1)

m−1

X

v=1

∆(vβ_v)

v

X

r=1

|tr|

r +O(1)mβ_m

m

X

v=1

|tv| v

=O(1)

m−1

X

v=1

|∆(vβ_v)|X_v+O(1)mβ_mX_m

=O(1)

m−1

X

v=1

|(v+ 1)∆β_v −β_v|X_v+O(1)mβ_mX_m

=O(1)

m−1

X

v=1

v|∆β_v|X_v +O(1)

m−1

X

v=1

|β_v|X_v+O(1)mβ_mX_m

=O(1) asm→ ∞, by (1.6), (1.10), (1.12), (2.2) and (2.3).

Again

m

X

n=1

1

n|T_n,2|=

m

X

n=1

|λ_n||t_n| n

=

m−1

X

n=1

∆|λ_n|

n

X

v=1

|t_v|

v +|λ_m|

m

X

n=1

|t_n| n

=O(1)

m−1

X

n=1

|∆λ_n|X_n+O(1)|λ_m|X_m

=O(1)

m−1

X

n=1

β_nX_n+O(1)|λ_m|X_m =O(1) asm→ ∞,

by (1.6), (1.7), (1.10) and (2.3). This completes the proof of the theorem.

REFERENCES

[1] Z.U. AHMAD, Absolute Nörlund summability factors of power series and Fourier series, Ann.

Polon. Math., 27 (1972), 9–20.

[2] S. ALJAN ˇCI ´C AND D. ARANDELOVI ´C, O-regularly varying functions, Publ. Inst. Math., 22 (1977), 5–22.

[3] H. BOR, Absolute Nörlund summability factors of power series and Fourier series, Ann. Polon.

Math., 56 (1991), 11–17.

[4] H. BOR, On the Absolute Nörlund summability factors, Math. Commun., 5 (2000), 143–147.

[5] M. FETEKE, Zur Theorie der divergenten Reihen, Math. es Termes Ertesitö (Budapest), 29 (1911), 719–726.

[6] N. KISHORE, On the absolute Nörlund summability factors, Riv. Math. Univ. Parma, 6 (1965), 129–134.

[7] E. KOGBENTLIANTZ, Sur lés series absolument sommables par la méthode des moyennes arith- métiques, Bull. Sci. Math., 49 (1925), 234–256.

[8] L. LEINDLER, A new application of quasi power increasing sequences, Publ. Math. Debrecen, 58 (2001), 791–796.

[9] F.M. MEARS, Some multiplication theorems for the Nörlund mean, Bull. Amer. Math. Soc., 41 (1935), 875–880.